Weebly · Web viewWe have the properties of equality, which helped us when solving algebraic proofs. Now, as we embark on geometric proofs, we will use some additional postulates

Chapter 6: 2-Column Proofs and CongruenceProperties of Equality and Congruence (6.1)

Segment/Angle Addition Postulate

Algebraic Proofs

GIVEN: d=ca=cd+p=na+c=n

WE CAN CONCLUDE:

Chapter 6: 2-Column Proofs and Congruence

Geometric Proofs

Chapter 6: 2-Column Proofs and CongruenceWe have the properties of equality, which helped us when solving algebraic proofs. Now, as we embark on geometric proofs, we will use some additional postulates and definitions:

Angle Addition Postulate:

Segment Addition Postulate:

Definition of midpoint:

Definition of bisector:

Definition of congruent:


6.3 Congruent Polygons and CirclesVocabulary:Congruent circles

Congruent polygons

Corresponding parts

Objectives:Define congruent polygons and congruent circles.Use correct notation and criteria for congruent polygons.

Congruent triangles


6.4 Parallel Lines and TransversalsObjectives:Identify transversalsName the special pairs of angles that form when a transversal intersects two linesIdentify the congruent pairs of angle formed by transversals through parallel lines.

Transversal:

Alternate interior angles

Alternate exterior angles

Corresponding angles

Same side interior angles

Congruent sides: Congruent angles: Congruent triangles:

Congruent sides: Congruent angles: Congruent triangles:


Same side exterior angles

Parallel postulate: Two lines intersected by a transversal are parallel if and only if the alternate interior angles are congruent.

Alternate exterior angles theorem: Two lines intersected by a transversal are parallel if and only if the alternate exterior angles are congruent.

Corresponding angles theorem: Two lines intersected by a transversal are parallel if and only if the corresponding angles are congruent.

Same side interior angles theorem: Two lines intersected by a transversal are parallel if and only if the same side interior angles are supplementary.

Same side exterior angles theorem: Two lines intersected by a transversal are parallel if and only if the same side exterior angles are supplementary.

Chapter 6: 2-Column Proofs and CongruenceAlternate Exterior Angles Theorem (1): If two parallel lines are intersected by a transversal, then the alternate exterior angles are congruent.

Alternate Exterior Angles Theorem (2): If a transversal intersects two lines such that the alternate exterior angles are congruent, then the lines are parallel.

Alternate Interior Angles Theorem (1): If two parallel lines are intersected by a transversal, then the alternate interior angles are congruent.

Alternate Interior Angles Theorem (2): If a transversal intersects two lines such that the alternate interior angles are congruent, then the lines are parallel.

Chapter 6: 2-Column Proofs and CongruenceCorresponding Angles Theorem (1): If a transversal intersects two lines such that the corresponding angles are congruent, then the two lines are parallel.

Corresponding Angles Theorem (2): If two lines are parallel, then the corresponding angles are congruent.

Proofs using Transversals

Simple Proofs With Triangles


6.5 Angles of PolygonsVocabularyIncluded Sides

Included Angles

Objectives:Prove theorems about the angles of trianglesDevelop formulas concerning the interior angles of polygons

Prove: The sum of the measures of the angles of any triangle is 180 degrees.

Prove: If two angles of one triangle are congruent to two angles of another triangle, then the third angles are also congruent.


Prove: The acute angles of a right triangle are complementary.

Angles in polygons:

*multiply the number of triangles by 180 to get the degrees in the polygon (180*t).*to get the measure of each angle, divide the total degrees by the number of interior angles of the polygon.

Find the measure of each angle of a hexagon:

6.6 ASA, SASObjectives:Identify and use ASA, SAS congruencies for triangles in proofs

SAS Congruence Postulate:

ASA Congruence Postulate:



6.7 AASObjectives:Identify and use AAS congruencies for triangles in proofs

SAA (AAS) Congruence Theorem:

Isosceles Triangles:



6.8 SSSObjectives:Identify and use SSS congruencies for triangles in proofs

SSS Congruence Theorem:

*** NO ASS, and no AAA in Geometry!


Documents

Weebly · Web viewWe have the properties of equality, which helped us when solving algebraic proofs. Now, as we embark on geometric proofs, we will use some additional postulates